Jordan decomposition/Levi decomposition in GL(n) in positive characteristic

Question

Let $k$ be a non archimedean field of positive characteristic. Lets consider a parabolic subgroup $P \subset GL(n, k)$.

I am a little bit confused by the following statement in "Laumon - Cohomology of Drinfeld Modular ... ":

I have an issue with the following two assertions

$P = MN$ has a Levi decomposition over $k$ (pg.123)

and

$\gamma \in P$ can be written as $\gamma = \gamma_m \gamma_n$ with $\gamma_m \in M$ and $\gamma_n \in N$ (pg.124)

Now, I have read that the Jordan decomposition and Levi decomposition need not to hold in positive characteristic (e.g. in Humpreys, Waterhouse). Do they mean that the decomposition are not functorial with respect to field extensions, and are available for the group, but not the group scheme?

Why is this not a contradiction?

Remark: I understand that elliptic element can become unipotent in an algebraic extension, since the minimal polynomial might not be separable in general.

Answer 1

Both those assertions describe a Levi decomposition. In general, groups need not have Levi decompositions, but parabolic subgroups of reductive groups do. This is proven for connected reductive groups (e.g. $GL(n, k)$) in Borel "Linear Algebraic Groups": see 20.5 for the decomposition over k.